• Characterisations. Let $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ be a relation. The following conditions are equivalent:
    1. There exists a function $f\colon A\to B$ such that $R=\text{Gr}\webleft (f\webright )$.
    2. The relation $R$ is total and functional.
    3. The weak and strong inverse images of $R$ agree, i.e. we have $R^{-1}=R_{-1}$.
    4. The relation $R$ has a right adjoint $R^{\dagger }$ in $\mathrm{Rel}$.

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